Integrating the Yukawa Potential: Understanding the 3rd Step

[Breo]

Please, can anyone explain me the steps made in the resolution of this integral?

http://en.wikipedia.org/wiki/Common...wa_Potential:_The_Coulomb_potential_with_mass

 

[ChrisVer]

1st step: go to "spherical coordinates" and integrating out the $\phi$... $u = \cos \theta$
2nd step: Integrating wrt to $u$, will give a difference of $e^{i~something} - e^{-i~something} \propto \sin (something)$. In fact I wouldn't ever write it in terms of sin...
3rd step: uses that he has two integrals one with the exponential with + and the other with the exponential with - ... then changes the integration variable of the one from k to -k, and gets this result.
4th step: factorizes the denominator.
5th step and then final: solves the (complex) integral

If you want to see a step in more details, you can ask for a specific one.

 

[Breo]

I actually get it with your points! Thank you very much!

 

[Muh. Fauzi M.]

1st step: go to "spherical coordinates" and integrating out the $\phi$... $u = \cos \theta$
2nd step: Integrating wrt to $u$, will give a difference of $e^{i~something} - e^{-i~something} \propto \sin (something)$. In fact I wouldn't ever write it in terms of sin...
3rd step: uses that he has two integrals one with the exponential with + and the other with the exponential with - ... then changes the integration variable of the one from k to -k, and gets this result.
4th step: factorizes the denominator.
5th step and then final: solves the (complex) integral

If you want to see a step in more details, you can ask for a specific one.

hello sir,
i can't get the 3rd step. is change the integration variable, change the value from k to -k in that term, including the dk -> d(-k)?

 

[ChrisVer]

i can't get the 3rd step. is change the integration variable, change the value from k to -k in that term, including the dk -> d(-k)?

like everytime you change your integration variable from $x$ to $y(x)$ what has to change is of course the differential, the integrand and the limits of the integral.